A gargosian cleric and a Tempussian priest meet, are they required by the tenets of there faith to do battle, or is it just don't work with them? I read up on Garagos and he hates Tempus pretty bad, but Garagos I imagine hates anything he can't kill, so I'd figure yes, but what about Tempus?

Same for Red Knight vs Garagos, is it autmatic hate, or just ignore me, I ignore you?

Its not automatic, no.

The clergies of Tempus and Garagos would certainly be in conflict generally, however.

Unless something very odd occurs, a cleric of Tempus and a cleric of Garagos would never be BFF.

They'd be certainly rivals though, right?

yes